A small particle of mass m has potential energy U = (1/2)mω²r², where ω is constant and r is distance from origin. Assuming Bohr quantization and a circular orbit, the radius of the nth orbit is proportional to:
Options

Question

A small particle of mass m has potential energy U = (1/2)mω²r², where ω is constant and r is distance from origin. Assuming Bohr quantization and a circular orbit, the radius of the nth orbit is proportional to:

Options

G1 Physics · Accepted Answer

For the circular orbit in the potential U = (1/2)mω²r², the radial force is F = −dU/dr = −mω²r For a circular orbit: mv²/r = mω²r Therefore, v = ωr Bohr quantization gives mvr = nħ So, m(ωr)r = nħ mωr² = nħ Hence, r² ∝ n and therefore r ∝ √n Hence option D follows.